Symmetry in 2D 1
Outlook Symmetry: definitions, unit cell choice Symmetry operations in 2D Symmetry combinations Plane Point groups Plane (space) groups Finding the plane group: examples 2
Symmetry Symmetry is the preservation of form and configuration across a point, a line, or a plane. The techniques that are used to "take a shape and match it exactly to another are called transformations Inorganic crystals usually have the shape which reflects their internal symmetry 3
Lattice = an array of points repeating periodically in space (2D or 3D). Motif/Basis = the repeating unit of a pattern (ex. an atom, a group of atoms, a molecule etc.) Unit cell = The smallest repetitive volume of the crystal, which when stacked together with replication reproduces the whole crystal 4
Unit cell convention By convention the unit cell is chosen so that it is as small as possible while reflecting the full symmetry of the lattice (b) to (e) correct unit cell: choice of origin is arbitrary but the cells should be identical; (f) incorrect unit cell: not permissible to isolate unit cells from each other (1 and 2 are not identical) 5 A. West: Solid state chemistry and its applications
Some Definitions Symmetry element: An imaginary geometric entity (line, point, plane) about which a symmetry operation takes place Symmetry Operation: a permutation of atoms such that an object (molecule or crystal) is transformed into a state indistinguishable from the starting state Invariant point: point that maps onto itself Asymmetric unit: The minimum unit from which the structure can be generated by symmetry operations 6
From molecular point group to space groups Complete consideration of all symmetry elements and translation yields to the space groups benzene graphene graphite D6h or 6/mmm Point group p6mm Plane group = point group symmetry + in plane translation P6 3 /mmc Space group = point group symmetry + in 3D translation 7
Symmetry operations in 2D*: 1. translation 2. rotations 3. reflections 4. glide reflections Symmetry operations in 3D: the same as in 2D + inversion center, rotoinversions and screw axes * Besides identity 8
1. Translation ( move ) Translation moves all the points in the asymmetric unit the same distance in the same direction. There are no invariant points (points that map onto themselves) under a translation. Translation has no effect on the chirality of figures in the plane. 9
2. Rotations A rotation turns all the points in the asymmetric unit around one axis, the center of rotation. The center of rotation is the only invariant point. A rotation does not change the chirality of figures. 10
Symbols for symmetry axes Drawn symbol One fold rotation axis two fold rotation axis --- (monad) (diad) Axes perpendicular to the plane Axes parallel to the plane three fold rotation axis four fold rotation axis six fold rotation axis (triad) (Tetrad) (Hexad) CRYSTALS MOLECULES 11
3. Reflections A reflection flips all points in the asymmetric unit over a line called mirror. The points along the mirror line are all invariant points A reflection changes the chirality of any figures in the asymmetric unit Symbol: m Representation: a solid line 12
4. Glide Reflections Glide reflection reflects the asymmetric unit across a mirror and then translates it parallel to the mirror There are no invariant points under a glide reflection. A glide plane changes the chirality of figures in the asymmetric unit. Symbol: g Representation: a dashed line 13
Point group symmetry Point group = the collection of symmetry elements of an isolated shape Point group symmetry does not consider translation! The symmetry operations must leave every point in the lattice identical therefore the lattice symmetry is also described as the lattice point symmetry Plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern 14
Examples of plane symmetry in architecture 15
Crystallographic plane point groups = 10 1. 1 (one fold axis) 6. m (mirror line) 2. 2 (two fold axis) 7. 2 mm (two mirror lines and a 2-fold axis)* 3. 3 (three fold axis) 8. 3 m (one 3-fold axis and three mirror lines) 4. 4 (four fold axis) 9. 4 mm (4-fold axis and four mirror lines)* 5. 6 (six fold axis) 10. 6 mm (6-fold axis and 6 mirror lines)* * Second m in the symbol refers to the second type of mirror line 16
5-fold, 7-fold, etc. axes are not compatible with translation non-periodic two dimensional patterns Ex: Starfish Non-periodic 2D patterns 5m (five fold axis + mirror) Wikipedia.org A Penrose tiling Group of atoms or viruses can form quasicrystals (quasicristals = ordered structural forms that are non-periodic) Electron diffraction of a Al-Mn quasicrystal showing 5-fold symmetry by Dan Shechtman 17
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html 18
Combining symmetry operations Ten different plane point groups : 1, 2, 3, 4, 6, m, 2 mm, 3 m, 4 mm, 6 mm Five different cell lattice types: 1. oblique(parallelogram) (a b, 90 ) 2. Rectangular (a b, 90ᵒ) 3. Square (a = b, 90ᵒ) 4. Centered rectangular or diamond (a b, 90ᵒ) 5. Rhombic or hexagonal (a = b, 120ᵒ) When point group symmetries are combined with the possible lattice cells 17 plane groups. 19
1. Combining rotation with translation 1. The rotations will always be to the plane (the space in 2D) 2. An -fold rotation followed by translation to it gives another rotation of the same angle (same order), in the same sense 3. The new rotation will be located at a distance x = T/2 x cotg /2 along perpendicular bisector of T (T=cell edge translation) Ex: 2-fold rotation followed by translation ( =180 ) 180 A 1 B x T 2 3 is the motif The second rotation will be on T in the middle at B Steps: 1. 2-fold rotation through A moves the motif from 1 to 2 2. translation by T moves the motif from 2 to 3 Or 1. 2-fold rotation through B moves the motif from 1 to 3 20
Pair of motifs: 2-fold axis combined with translation 1 8 4 T2 T1 T1 T1+T2 6 7 1 4 2 T2 9 3 2 3 2-fold rotation at 1 combined with translation T 1 gives the rotation 6 (rotation 6 is translated to 7 by T2) 2-fold rotation at 1 combined with translation T 2 gives the rotation 8 (rotation 8 is translated to 9 by T1) 2-fold rotation at 1 combined with translation T 1+T2 gives the rotation in the middle The blue, red, green and yellow marked are independent 2-fold axes: they relate different objects pair-wise in the pattern no any pair of the blue and one of the red, green or yellow 2-fold axis describe the same pair-wise relationship 21
6-fold axis combined with translation 6-fold axis contains 2 /6, 2 /3, 2 /2 rotations All the operations of a 3-fold axis combined with translation and of a 2-fold axis combined with translation will be included for a p6 plane group 22
Combination of the rotation axes with a plane lattice = translation Two fold axis Three fold axis Four non-equivalent 2-fold axes to the plane (0 0; ½ ½, ½ 0, 0 ½ ) Four fold axis Three non-equivalent 3-fold axes to the plane 00, 2 / 3 1 / 3, 1 / 3 2 / 3 ) Six fold axis Two non-equivalent 4-fold axes to the plane; One non-equivalent 2-fold axis to the plane; (00, ½ ½) and ( ½ 0, 0 ½ ) Martin Buerger: An introduction to fundamental geometric features of crystals One non-equivalent 6-fold axes to the plane; One non-equivalent 3-fold axis to the plane; (0 0) ; ( 2 / 3 1 / 3, 1 / 3 2 / 3 ) and ( ½ 0) 23
2. Combining a reflection with translation A reflection combined with a translation to it is another reflection at ½ of that perpendicular translation 1. A rectangular cell 1 2 2 1 3 - Pair of motifs 1 1 2 Translatio n 3 * 2 *the mirror 2 is situated at ½ distance of the translation The mirror 2 is independent from 1 because the position of the objects (1 and 2) relative to the mirror in the center ( 2 )of the cell is distinct from the position of the same objects relative to the first mirror ( 1 ) 24
2. A centered rectangular cell 1 and 2 are equivalent because we must have a motif in the center A glide line results in here - Pair of motifs 2 1 1 2 2 2 2 1 1 1 1 2 A glide is the result of a reflection and a translation 1 T T *T(T +T )=glide plane The glide will be at the half distance of T T 25
3. Combining a glide with a translation 1. A rectangular cell gliding 1 2 Translatio n 3 2 g 1 1 g 2 3 gliding by g 2 The glide g 2 is situated at half of the translation which is perpendicular to it - Motif 2 g1 1 T( ) g2 3 3 Reflecting 1 by a mirror in the center of the edges gives 3 ; Gliding 3 half of T parallel gives 3 26
2. A centered rectangular cell Combining a glide plane with a translation in a centered rectangular lattice gives a mirror plane situated at ½ of T/2. 2 g1 1 g2 27
4. Combining two reflections The operation of applying two reflections in which the mirror planes ( 1 and 2 ) are making an angle with each other is the same with the rotation by an 2 angle 1 1 1 2 2 Guide to the eye Two reflections: 1 1 by reflection on 1 1 2 by reflection on 2 1 2 1 1' 2 One rotation: 1 2 by two times rotation rotation by 2 28
5. Combining a rotation with a reflection A rotation by followed by a reflection 1 will result in another reflection which will be situated at an angle /2 relative to the first reflection 2 1 3 3 1 1 2 2 1 1 rotationby reflection 2 by 3 reflection by 2 29
Combining symmetry operations 1. Oblique (parallelogram) (a b, 90 ) Plane groups p1 and p2 p stands for the fact that we have only one lattice point per cell primitive lattice p1 p2 Examples of motifs having point group 1: (The motif itself should have no symmetry) and Examples of motifs having point group 2: and (The motif itself should have a 2-fold axis) 30
Plane group symbol rules/meaning 1. First letter: p or c translation symmetry + type of centering 2. The orientation of the symmetry elements: to coordinate system x, y and z. The highest multiplicity axis or if only one symmetry axis present they are on z Ex: p4mm: 4-fold axis in the z direction; p3m1: 3-fold axis in the z direction The highest symmetry axis is mentioned first and the rest are omitted ex: p4mmm: 4-fold axis on z and two 2-fold axes are omitted If highest multiplicity axis is 2-fold the sequence is x-y-z ex: pmm2; pgm2; cmm2: 2-fold axis on z 3. The addition of 1 is often used as a place holder to ensure the mirror or glide line is correctly placed ex: p3m1 and p31m 4/24/2013 L. Viciu AC II Symmetry in 2D m y m x m z 31
2. Rectangular (a b, 90ᵒ) Plane groups: pm, pg, pmg2, pmm2 and pgg2 pmg2 pgg2 pmm2 Possible motifs: 4/24/2013 m L. Viciu AC II Symmetry in 2D 2mm 32
2. Examples of Rectangular plane groups with glide lines motif: motif: pmg2 pgg2 pmg2 pgg2 33
3. Square (a = b, 90ᵒ) Plane groups: p4, p4mm and p4gm Possible motifs: 4 4mm 34
Questions to recognize a square plane group 1. Is there a 4-fold axis? It should be otherwise it cannot be a square lattice 2. Is there a mirror line in there? If No, then is a p4 plane group If Yes, 3. Is the mirror line passing through a 4-fold axis? If Yes then the plane group is p4mm If No then the group is a p4mg 35
4. Centered rectangular (a b, 90ᵒ) The dash lined cell is known as diamond or rhombus cell Plane groups: cm and cmm2 Possible motifs: cmm2 m 2mm 36
Diamond vs. centered rectangular The diamond lattice has a mirror through it such that always a = b but the angle is general a a=b The centered rectangular lattice has now 2 atoms per unit cell The centered rectangular lattice has 2-fold redundancy (two diamond unit cells) but it has the big advantage of an orthogonal coordinate system. Therefore it is the standard cell 37
5. Rhombic or hexagonal (a = b, 120ᵒ) Plane groups: p3, p31m, p3m1, p6 and p6mm Possible motifs: 6 6mm 3 3m 38
How the motifs are oriented in p3m plane group p3m1 The mirrors are to the translation (the translation comes in the middle of the mirrors) p31m The translation is along the mirror planes On the second place in the plane group symbol comes what is to the cell edge and on the third place comes what is to the cell edge 4/23/2013 L. Viciu AC II Symmetry in 2D 39
When we have translations which are inclined to the mirrors like in p3m1 plane group, a glide is always interleaved between the two mirrors The glide is parallel to the mirrors at half distance between them 1 2 a) the inclination of translation relative to the mirrors b) the location of glide (between the mirrors at the half distance) 4/23/2013 L. Viciu AC II Symmetry in 2D 40
When we have translations which are inclined to the mirrors like in p31m plane group, a glide is always interleaved between the two mirrors. The glide is parallel to the mirrors at half distance between them. a) The inclination of the translation relative to the mirrors b) The location of the glides (between the mirrors at the half distance) 4/23/2013 L. Viciu AC II Symmetry in 2D 41
The p6mm plane group has the symmetry elements of both p3m1 and p31m groups because both of these groups are present simultaneously in p6mm plane group. p3m1 +p31m When we add the symmetry elements we should make sure that all the symmetry elements are left invariant (we don t create additional translations or consequently more axes and planes; 4/23/2013 L. Viciu AC II Symmetry in 2D 42
Symmetry Elements of the 2D Space Groups 4/23/2013 Unit cell edge mirror line L. Viciu AC II Symmetry in 2D glide line 2, 3, 4, 6 fold axes 43
The equivalence of atom positions results from translation y x y x The atom will be then moved by translation to every lattice point y x The atom at the lattice point has the coordinates: (x, y) The 2 fold axes place the atoms at the opposite direction It is possible to say also 1-x 1-y But is more esthetic to give the positions x y and x y y x 1-x 1-y 4/23/2013 L. Viciu AC II Symmetry in 2D 44
1. Highest order rotation? Yes 2. Has reflection? 6-fold p6mm p6 4-fold 3-fold 2-fold Yes: p4mm 3. Has mirrors at 45? No: p4gm 3. Has rot. centre off mirrors? Yes: p31m No: p3m1 3. Has perpendicular reflections? Yes Has rot. centre off mirrors? Yes: cmm2 No: pmm2 No No p4 p3 Has glide reflection? pmg2 Yes: pgg2 No: p2 none Has glide axis off mirrors? Has glide reflection? Yes: cm No: pm Yes: pg No: p1 4/23/2013 L. Viciu AC II Symmetry in 2D 45
Fundamental Steps in Plane Groups Identification 1. Locate the motif present in the pattern. This can be a molecule, molecules, atom, group of atoms, a shape or group of shapes. The motif can usually be discovered by noting the periodicity of the pattern. 2. Identify any symmetry elements in the motif. 3. Locate a single lattice point for each occurrence of the motif. It is a good idea to locate the lattice points at a symmetry element location. 4. Connect the lattice points to form the unit cell. 5. Determine the plane group by comparing the symmetry elements present to the 17 plane patterns. 4/23/2013 L. Viciu AC II Symmetry in 2D 46
Finding the plane group No symmetry besides translation: The lattice type is oblique, plane group p1. Each unit mesh (unit cell) contains 1 white bird and 1 blue bird. 4/23/2013 L. Viciu AC II Symmetry in 2D 47
Finding the plane group No symmetry besides translation: The lattice type is oblique, plane group p1. Each unit mesh (unit cell) contains 1 white bird and 1 blue bird. 4/23/2013 L. Viciu AC II Symmetry in 2D 48
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 49
Finding the plane group 1. Highest order rotation? A: 2 2. Has reflections? A: yes 3. Has rotation centers off mirrors? A: yes 4. Space group: A: cmm2 4/23/2013 L. Viciu AC II Symmetry in 2D 50
Finding the plane group The unit cell is square. Symmetry elements: -2-fold axis -Two mirror lines ( to each other) - Two glide lines Plane group: cmm2 4/23/2013 L. Viciu AC II Symmetry in 2D 51
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 52
Finding the plane group 1. Highest order rotation? A: 3 2. Has reflections? A: yes 3. Has rotation centers off mirrors? A: No 4. Space group: A: p3m1 4/23/2013 L. Viciu AC II Symmetry in 2D 53
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 54
Finding the plane group 1. Highest order rotation? A: 6 2. Has reflections? A: yes 3. Space group: A: p6mm 4/23/2013 L. Viciu AC II Symmetry in 2D 55
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D Christopher Hammond: The basics of crystallography and diffraction (third edition) 56
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D Christopher Hammond: The basics of crystallography and diffraction (third edition) 57
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 58
p4gm 4/23/2013 L. Viciu AC II Symmetry in 2D 59
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 60
Finding the plane group 4/23/2013 L. Viciu AC II Symmetry in 2D 61